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Primitive of a function

Calculation of `\int f(x) \ dx`
Enter a single letter
To multiply: write a*b not ab

This tool calculates the primitive of a function.
Usual functions are accepted: sine, cosine, tangent, logarithm (log), exponential, root, etc. (See table below).

How to use this calculator ?

Variables A function can have one or more variables, but only one main variable.
A variable is a single lowercase or uppercase letter.
Examples:
A function f with one main variable : f(x) = 4*x
A function g with one main variable x and a secondary parameter m,
g(x) = 4*x*m + x + 1,
In this case, enter x in the “main variable” field
Numbers Use a dot as decimal separator
Operators + (addition),
- (substration),
* (multiplication),
/ (division),
^ (power),
For multiply operator, enter a*b not a.b nor ab. Example: 2*x.
Constants You may use these constants :
pi (approx. 3.14),
e (approx. 2,72)
Examples: f(x) = pi * x or f(x) = e * (x+1+2*e)2
Common Functions You may use theses functions in the expression of f(x)
sqrt(x) (square root),
exp(x) (exponential function),
log(x) or ln (natural logarithm),
Trigonometric functions You may use theses functions in the expression of f(x)
sin (sine),
cos (cosine),
tan (tangent),
cot (cotangent),
sec (secant),
csc (cosecant),
Inverse trigonometric functions You may use theses functions in the expression of f(x)
arcsin (arcsine),
arccos (arccosine),
arctan (arctangent),
arccot (arcotangent),
arcsec (arcsecant),
arccsc (arccosecant),
Hyperbolic Functions You may use theses functions in the expression of f(x)
sinh (hyperbolic sine),
cosh (hyperbolic cosine),
tanh (hyperbolic tangent),
coth (hyperbolic cotangent),
sech (hyperbolic secante),
csch (hyperbolic cosecant)
Inverse hyperbolic functions You may use theses functions in the expression of f(x)
asinh (inverse hyperbolic sine),
acosh (inverse hyperbolic cosine),
atanh (inverse hyperbolic tangent),
acoth (inverse hyperbolic cotangent),
asech (inverse hyperbolic secant),
acsch (inverse hyperbolic cosecant)

Table of basic functions primitives

Fonction f(x)Primitive
`k` where `k in RR``k*x+C`
`x^n` where `n in NN`*`x^(n+1)/(n+1)+C`
`1/x``ln(|x|)+C`
`1/x^n` where `n in NN , n >=2``-1/((n-1)x^(n-1))+C`
`sqrt(x)``frac{2}{3}*x*sqrt(x)+C`
`1/sqrt(x)``-1/(2*x*sqrt(x))+C`
`sin(x)``-cos(x)+C`
`cos(x)``sin(x)+C`
`ln(x)``x*ln(x)-x+C`
`e^x``e^x+C`

Table of composite functions primitives

Fonction composéePrimitive
`u'*u``u^2/2+C`
`(u')/u^2``-1/u+C`
`u'*u^n` where `n in NN\text{ and }n != -1``u^(n+1)/(n+1)+C`
`(u')/u^n` where `n in NN\text{ and }n >= 2``1/((n-1)*u^(n-1))+C`
`(u')/sqrt(u)``2*sqrt(u)+C`
`(u')/u``ln(|u|)+C`
`u'*e^u``e^u+C`
`u'*sin(u)``-cos(u)+C`
`u'*cos(u)``sin(u)+C`
`u'*u^a` where `a in RR\text{ and }a != -1``u^(a+1)/(a+1)+C`
`u'*g(u)``g(u)+C`

See also

Derivative calculator
Taylor series expansion
Function limit calculator
Value of a function
Definite Integral